EN
We study the class of modules which are invariant under idempotents of their envelopes. We say that a module M is 𝓧-idempotent-invariant if there exists an 𝓧-envelope u : M → X such that for any idempotent g ∈ End(X) there exists an endomorphism f : M → M such that uf = gu. The properties of this class of modules are discussed. We prove that M is 𝓧-idempotent-invariant if and only if for every decomposition $X = ⨁ _{i∈ I}X_{i}$, we have $M = ⨁ _{i∈ I} (u^{-1}(X_{i}) ∩ M)$. Moreover, some generalizations of 𝓧-idempotent-invariant modules are considered.